The Jealous Neighbors

Mar 03, 2018

RAY: This week's Puzzler was sent to me from Bruce Robinson, a professor of civil and environmental engineering at the University of Tennessee. I think he sent it some time in 1980.

TOM: He's probably quit or retired by now.

RAY: There are 25 jealous people who live in the squares of a five-by-five grid. We're gonna number the squares, starting in the upper left-hand corner, 1 through 25.

TOM: So the first row starts with 1, the second row starts with 6, the third row starts with 11, and so forth.

RAY: Right. Remember, each person is jealous of his adjacent neighbor. Not his diagonal neighbor, but the person up or down or left or right of him. Each aspires to move into the apartment of his adjacent neighbor.

The question is very simple: What is the fewest number of total moves that can accomplish this?

Answer:

RAY: So, if you draw this grid, the square in the upper left-hand corner we could say is one, and the one next to it is two, three, four, five, and then the line below that is six, seven, eight, nine, 10, 11, 12, right? All the way to 25. Got it?

TOM: I got it.

RAY: Now, each person who lives on the floor aspires to move into the apartment of one of his adjacent neighbors. So number one can move to square number two, or number six, for example.

TOM: But not diagonally.

RAY: Not diagonally. Number two can move to number one.

TOM: Or number three, or number seven.

RAY: There you go.

TOM: I can visualize all that.

RAY: So, here's the question. Why would anyone live in such a stupid building? No, the question is, what is the fewest number of total moves that will allow every person to move to an adjacent square.

TOM: All right. I see --

RAY: I know Doug Mayer has been, did you get the answer, Mayer? He says, no, I didn't get the answer.

TOM: Well, unencumbered by the thought process, I concluded right away when you gave this problem --

RAY: It had to be one or zero, right?

TOM: -- that the answer was, it was either going to be 26 --

RAY: Ah! Very good!

TOM: -- or millions.

RAY: Well, millions is close.

TOM: Millions is closer, huh?

RAY: If you don't number them one through 25, but instead, letter them.

TOM: Yeah.

RAY: And not A, B, C, D, E. Let's letter the first one A.

TOM: Yeah.

RAY: The next one B, the next one A, the next one B, et cetera, et cetera.

TOM: Oh, oh, oh.

RAY: Then, everyone who's on an A square must, by definition, move to what?

TOM: A B square.

RAY: Right.

TOM: And then, vice versa.

RAY: And everyone who's on a B square must move to an A square. It's pretty obvious if you draw it out.

TOM: Yeah. That's true.

RAY: Yeah. Now, if you add them up, by some stroke of bad luck, you got 13 A squares and only 12 B squares.

TOM: Someone's got to move out of the building.

RAY: They've got them mixed up, like. So, there are no fewest number of moves. It is impossible for this to happen. I know, it was a little sneaky.

TOM: It's impossible!

RAY: Poor Mayer just threw a waste basket at me! He was up to 26,215 moves, and he almost had it!